I have looked for properties on auction sites and the traditional real estate sites.

You should be able to get more than two loans still traditionally!

Property that has been purchased with the primary intent of immediate resale (yes, fix n flips as they are traditionally understood) does not qualify for1031 treatment.  

OK - I'm a newbie here, so pardon any ignorance on terms or how stuff works...I've hit a brick wall trying to refi my current rental to get cash out for my next one (ugh.) So, I'm thinking maybe I need to switch gears...

A traditional mortgage can be used to finance it if it is your own name.

The letter from Doctor shows the animal as the "medicine," the verification request form confirms the need of the animal.

Again, I have a completely different rental model than the traditional buy and hold so take that as you will.I would argue that the price of the property is too high for the rent he is getting.

Now one is left with the possible cash flow in a traditionally and typically non appreciating market and asset.

If you find a TK provider that isn't willing to work with traditional lenders, requires cash investment or pushes hard money lending even for inexperienced investors, then that might be a red flag.... 

You may also be able to invest in vacation rentals and convert a traditional SFR into an Airbnb type propert.